Kodai Mathematical Journal

On holomorphic families of rational maps: finiteness, rigidity and stability

Hiroshige Shiga

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Abstract

We consider holomorphic families of rational maps from the viewpoint of complex dynamics.

First, we consider some classes of families of rational maps which satisfy a certain stability condition. We show a finiteness theorem for such holomorphic families of rational maps parameterized by a Riemann surface of finite type.

Next, we consider the monodromy of quasiconformally stable holomorphic families of rational maps over a punctured disk, and study the action of the monodromy on the Julia set.

Article information

Source
Kodai Math. J., Volume 24, Number 1 (2001), 48-65.

Dates
First available in Project Euclid: 19 January 2005

Permanent link to this document
https://projecteuclid.org/euclid.kmj/1106157295

Digital Object Identifier
doi:10.2996/kmj/1106157295

Mathematical Reviews number (MathSciNet)
MR1813718

Zentralblatt MATH identifier
0993.37025

Subjects
Primary: 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations
Secondary: 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04] 37F15: Expanding maps; hyperbolicity; structural stability

Citation

Shiga, Hiroshige. On holomorphic families of rational maps: finiteness, rigidity and stability. Kodai Math. J. 24 (2001), no. 1, 48--65. doi:10.2996/kmj/1106157295. https://projecteuclid.org/euclid.kmj/1106157295


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